DynBindModelExamples

Author: Matthias Konrad

The system

In this example we are studying a simple system with three classes that contains both renaming and selects:

class A feature f1 do ... end g1 do ... end end

class B inherit A rename f1 as f2, g1 as g2 redefine f2, g2 end feature f2 do ... end g2 do ... end end

class D inherit   A redefine f1, g1 select g1 end B select f2 end feature f1 do ... end g1 do ... end end

This system is defined by:

• $S=\{A,B,D\}$
• $C=\{1\mapsto A,2\mapsto B,3\mapsto D\}$
• $S_1=\left\{C_1\right\},S_2=\{C_1,C_2\},S_3=\{C_1,C_2,C_3\}$
• $\tau =\{C_1\mapsto \{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\},C_2\mapsto \{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\},C_3\mapsto \{f_1\mapsto \underset{\_}{C_3.f_1},g_1\mapsto \underset{\_}{C_3.g_1}\}\}$
• $\eta =\{C_2\mapsto \{C_1\mapsto \{f_1\mapsto f_2,g_1\mapsto g_2\}\}\}$
• $\alpha =\{C_2\mapsto \left\{C_1\right\},C_3\mapsto \{C_2,C_1\}\}$
• $\sigma =\{C_3\mapsto \{C_2\mapsto \left\{f_2\right\},C_1\mapsto \left\{g_1\right\}\}\}$

The naming function

We proceed by calculating the naming function. According to its definition we have to do that incrementally. Meaning that we first calculate ${\beta }_{S_1}$ then ${\beta }_{S_2}$ and finally ${\beta }_{S_3}$.

Calculating ${\beta }_{S_1}$

Since ${\beta }_{S_1}$ contains only one class it is sufficient to define ${\beta }_{S_1}(C_1\mapsto C_1)$. According to the definition:

• ${\beta }_{S_1}(C_1\mapsto C_1)=\text{id}(\text{dom}\left({\tau }_1\right)\cup \text{UNION}b\cdot (b\in {\alpha }_1|\text{ran}({\beta }_{S_0}(b\mapsto b)\ll {\eta }_{1,b})))$

We can forget about the right part of the union since $C_1$ has no base classes. So we need to calculate:

1. $\text{id}\left(\text{dom}\left({\tau }_1\right)\right)$
   1. ${\tau }_1=\tau (C_1)=\{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\}$
   2. $\text{dom}\left({\tau }_1\right)=\{f_1,g_1\}$
   3. $\text{id}\left(\text{dom}\left({\tau }_1\right)\right)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$

So we get:

${\beta }_{S_1}(C_1\mapsto C_1)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$

And hence:

${\beta }_{S_1}=\{(C_1\mapsto C_1)\mapsto \{f_1\mapsto f_1,g_1\mapsto g_1\}\}$

Calculating ${\beta }_{S_2}$

We show the calculation of the different (four) parts of the function in separate sections:

Calculate ${\beta }_{S_2}(C_1\mapsto C_1)$

According to the definition: ${\beta }_{S_2}(C_1\mapsto C_1)={\beta }_{S_1}(C_1\mapsto C_1)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$

Calculate ${\beta }_{S_2}(C_2\mapsto C_1)$

According to the definition: ${\beta }_{S_2}(C_2\mapsto C_1)=\mathrm{\varnothing }$

Calculate ${\beta }_{S_2}(C_2\mapsto C_2)$

According to the definition: ${\beta }_{S_2}(C_2\mapsto C_2)=\text{id}(\text{dom}\left({\tau }_2\right)\cup \text{UNION}b\cdot (b\in {\alpha }_2|\text{ran}({\beta }_{S_1}(b\mapsto b)\ll {\eta }_{2,b})))$

Lets calculate the left part first:

1. ${\tau }_2=\tau (C_2)=\{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}$
2. $\text{dom}\left({\tau }_2\right)=\{f_2,g_2\}$

And now the right part:

1. $\text{UNION}b\cdot (b\in \left\{C_1\right\}|\text{ran}({\beta }_{S_1}(b\mapsto b)\ll {\eta }_{2,b}))=\text{ran}({\beta }_{S_1}(C_1\mapsto C_1)\ll {\eta }_{2,C_1})$
2. ${\beta }_{S_1}(C_1\mapsto C_1)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$
3. ${\eta }_{2,C_1}=\eta (C_2)(C_1)=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
4. $\{f_1\mapsto f_1,g_1\mapsto g_1\}\ll \{f_1\mapsto f_2,g_1\mapsto g_2\}=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
5. $\text{ran}\left(\{f_1\mapsto f_2,g_1\mapsto g_2\}\right)=\{f_2,g_2\}$

So we get:

${\beta }_{S_2}(C_2\mapsto C_2)=\{f_2\mapsto f_2,g_2\mapsto g_2\}$

Calculate ${\beta }_{S_2}(C_1\mapsto C_2)$

According to the definition: ${\beta }_{S_2}(C_1\mapsto C_2)={{\beta }_{S_2}}^{\prime }(C_1\mapsto C_2)\ll {\theta }_{S_2}(C_1\mapsto C_2)$

Where: ${{\beta }_{S_2}}^{\prime }(C_1\mapsto C_2)=\text{UNION}b\cdot (b\in {\alpha }_2|{\beta }_{S_1}(C_1\mapsto b);(\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto b)\right)\right)\ll {\eta }_{2,b}))$

And: ${\theta }_{S_2}(C_1\mapsto C_2)=\text{UNION}b\cdot (b\in {\alpha }_2|{\beta }_{S_1}(C_1\mapsto b);(\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto b)\right)\right)\ll {\eta }_{2,b})▹{\sigma }_{2,b})$

Here are the steps taken to compute the part:

1. ${\beta }_{S_1}(C_1\mapsto C_1);(\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto C_1)\right)\right)\ll {\eta }_{2,C_1})$
   1. $\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto C_1)\right)\right)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$
   2. $\{f_1\mapsto f_1,g_1\mapsto g_1\}\ll \{f_1\mapsto f_2,g_1\mapsto g_2\}=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
   3. $\{f_1\mapsto f_1,g_1\mapsto g_1\};\{f_1\mapsto f_2,g_1\mapsto g_2\}=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
   4. ${{\beta }_{S_2}}^{\prime }(C_1\mapsto C_2)=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
2. ${\beta }_{S_1}(C_1\mapsto C_1);(\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto C_1)\right)\right)\ll {\eta }_{2,C_1})▹{\sigma }_{2,C_1}$
   1. ${\sigma }_{2,C_1}=\mathrm{\varnothing }$
   2. $\text{id}\left(\text{ran}\left({\beta }_{S_1}(C_1\mapsto C_1)\right)\right)\ll \mathrm{\varnothing }=\mathrm{\varnothing }$
   3. ${\beta }_{S_1}(C_1\mapsto C_1);\mathrm{\varnothing }=\mathrm{\varnothing }$
   4. ${\theta }_{S_2}(C_1\mapsto C_2)=\mathrm{\varnothing }$
3. $\{f_1\mapsto f_2,g_1\mapsto g_2\}\ll \mathrm{\varnothing }=\{f_1\mapsto f_2,g_1\mapsto g_2\}$

So we get:

${\beta }_{S_2}(C_1\mapsto C_2)=\{f_1\mapsto f_2,g_1\mapsto g_2\}$

Putting the parts together:

We can know show the complete total function:

${\beta }_{S_2}=\{$	$(C_1\mapsto C_1)\mapsto \{f_1\mapsto f_1,g_1\mapsto g_1\},$
	$(C_1\mapsto C_2)\mapsto \{f_1\mapsto f_2,g_1\mapsto g_2\},$
	$(C_2\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_2\mapsto C_2)\mapsto \{f_2\mapsto f_2,g_2\mapsto g_2\}\}$

Calculating ${\beta }_{S_3}$

We will not show the complete construction of ${\beta }_{S_3}$ but restrict our self to the most interesting parts.

Calculate ${\beta }_{S_3}(C_1\mapsto C_3)$

According to the definition: ${\beta }_{S_3}(C_1\mapsto C_3)={{\beta }_{S_3}}^{\prime }(C_1\mapsto C_3)\ll {\theta }_{S_3}(C_1\mapsto C_3)$

Where: ${{\beta }_{S_3}}^{\prime }(C_1\mapsto C_3)=\text{UNION}b\cdot (b\in {\alpha }_3|{\beta }_{S_2}(C_1\mapsto b);(\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto b)\right)\right)\ll {\eta }_{3,b}))$

And: ${\theta }_{S_3}(C_1\mapsto C_3)=\text{UNION}b\cdot (b\in {\alpha }_3|{\beta }_{S_2}(C_1\mapsto b);(\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto b)\right)\right)\ll {\eta }_{3,b})▹{\sigma }_{3,b})$

Note that we can omit ${\eta }_{3,b}$ from the formulas since it is always empty.

1. ${\beta }_{S_2}(C_1\mapsto C_2);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_2)\right)\right)\cup {\beta }_{S_2}(C_1\mapsto C_1);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_1)\right)\right)$
   1. ${\beta }_{S_2}(C_1\mapsto C_2)=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
   2. $\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_2)\right)\right)=\{f_2\mapsto f_2,g_2\mapsto g_2\}$
   3. ${\beta }_{S_2}(C_1\mapsto C_2);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_2)\right)\right)=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
   4. ${\beta }_{S_2}(C_1\mapsto C_1)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$
   5. $\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_1)\right)\right)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$
   6. ${\beta }_{S_2}(C_1\mapsto C_1);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_1)\right)\right)=\{f_1\mapsto f_1,g_1\mapsto g_1\}$
   7. $\{f_1\mapsto f_2,g_1\mapsto g_2\}\cup \{f_1\mapsto f_1,g_1\mapsto g_1\}=\{f_1\mapsto f_1,f_1\mapsto f_2,g_1\mapsto g_1,g_1\mapsto g_2\}$
2. ${\beta }_{S_2}(C_1\mapsto C_2);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_2)\right)\right)▹{\sigma }_{3,C_2}\cup {\beta }_{S_2}(C_1\mapsto C_1);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_1)\right)\right)▹{\sigma }_{3,C_1}$
   1. ${\sigma }_{3,C_2}=\sigma \left(C_3\right)\left(C_2\right)=\left\{f_2\right\}$
   2. ${\sigma }_{3,C_1}=\sigma \left(C_3\right)\left(C_1\right)=\left\{g_1\right\}$
   3. ${\beta }_{S_2}(C_1\mapsto C_2);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_2)\right)\right)▹{\sigma }_{3,C_2}=\{f_1\mapsto f_2\}$
   4. ${\beta }_{S_2}(C_1\mapsto C_1);\text{id}\left(\text{ran}\left({\beta }_{S_2}(C_1\mapsto C_1)\right)\right)▹{\sigma }_{3,C_1}=\{g_1\mapsto g_1\}$
   5. $\{f_1\mapsto f_2\}\cup \{g_1\mapsto g_1\}=\{f_1\mapsto f_2,g_1\mapsto g_1\}$
   6. $\{f_1\mapsto f_1,f_1\mapsto f_2,g_1\mapsto g_1,g_1\mapsto g_2\}\ll \{f_1\mapsto f_2,g_1\mapsto g_1\}=\{f_1\mapsto f_2,g_1\mapsto g_1\}$

So we get:

${\beta }_{S_3}(C_1\mapsto C_3)=\{f_1\mapsto f_2,g_1\mapsto g_1\}$

The complete naming function

${\beta }_{S_3}=\{$	$(C_1\mapsto C_1)\mapsto \{f_1\mapsto f_1,g_1\mapsto g_1\},$
	$(C_1\mapsto C_2)\mapsto \{f_1\mapsto f_2,g_1\mapsto g_2\},$
	$(C_2\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_2\mapsto C_2)\mapsto \{f_2\mapsto f_2,g_2\mapsto g_2\},$
	$(C_1\mapsto C_3)\mapsto \{f_1\mapsto f_2,g_1\mapsto g_1\},$
	$(C_2\mapsto C_3)\mapsto \{f_2\mapsto f_2,g_2\mapsto g_2\},$
	$(C_3\mapsto C_3)\mapsto \{f_1\mapsto f_1,g_1\mapsto g_1,f_2\mapsto f_2,g_2\mapsto g_2\},$
	$(C_3\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_3\mapsto C_2)\mapsto \mathrm{\varnothing }\}$

The dynamic binding function

To calculate the dynamic binding function, we need to get ${\delta }_{S_3}$. We will again calculate it incrementally:

Calculating ${\delta }_{S_1}$

According to the definition:

${\delta }_{S_1}(C_1\mapsto C_1)={\beta }_S(C_1\mapsto C_1);((\text{UNION}b\cdot (b\in {\alpha }_1|{({\beta }_S(b\mapsto b);{\eta }_{1,b})}^{-1};{\delta }_{S_0}(b\mapsto b)))\ll {\tau }_1)$

Since ${\alpha }_1$ is empty this simplifies to: ${\delta }_{S_1}(C_1\mapsto C_1)={\beta }_S(C_1\mapsto C_1);{\tau }_1=\{f_1\mapsto f_1,g_1\mapsto g_1\};\{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\}$

So we get: ${\delta }_{S_1}=\{(C_1\mapsto C_1)\mapsto \{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\}\}$

Calculating ${\delta }_{S_2}$

Again we calculate the different parts of the function in separate sections:

Calculate ${\delta }_{S_2}(C_1\mapsto C_1)$

According to the definition: ${\delta }_{S_2}(C_1\mapsto C_1)={\delta }_{S_1}(C_1\mapsto C_1)=\{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\}$

Calculate ${\delta }_{S_2}(C_2\mapsto C_1)$

According to the definition: ${\delta }_{S_2}(C_1\mapsto C_1)=\mathrm{\varnothing }$

Calculate ${\delta }_{S_2}(C_2\mapsto C_2)$

According to the definition:

${\delta }_{S_2}(C_2\mapsto C_2)={\beta }_S(C_2\mapsto C_2);((\text{UNION}b\cdot (b\in {\alpha }_2|{({\beta }_S(b\mapsto b);{\eta }_{2,b})}^{-1};{\delta }_{S_1}(b\mapsto b)))\ll {\tau }_2)$

1. $\text{UNION}b\cdot (b\in \left\{C_1\right\}|{({\beta }_S(b\mapsto b);{\eta }_{2,b})}^{-1};{\delta }_{S_1}(b\mapsto b))={({\beta }_S(C_1\mapsto C_1);{\eta }_{2,C_1})}^{-1};{\delta }_{S_1}(C_1\mapsto C_1)$
   1. ${\beta }_S(C_1\mapsto C_1);{\eta }_{2,C_1}=\{f_1\mapsto f_2,g_1\mapsto g_2\}$
   2. ${({\beta }_S(C_1\mapsto C_1);{\eta }_{2,C_1})}^{-1}=\{f_2\mapsto f_1,g_2\mapsto g_1\}$
   3. ${({\beta }_S(C_1\mapsto C_1);{\eta }_{2,C_1})}^{-1};{\delta }_{S_1}(C_1\mapsto C_1)=\{f_2\mapsto \underset{\_}{C_1.f_1},g_2\mapsto \underset{\_}{C_1.g_1}\}$
2. ${\tau }_2=\tau \left(C_2\right)=\{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}$
3. $\{f_2\mapsto \underset{\_}{C_1.f_1},g_2\mapsto \underset{\_}{C_1.g_1}\}\ll \{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}=\{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}$

So we get: ${\delta }_{S_2}(C_2\mapsto C_2)=\{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}$

Calculate ${\delta }_{S_2}(C_1\mapsto C_2)$

According to the definition:

${\delta }_{S_2}(C_1\mapsto C_2)={\beta }_S(C_1\mapsto C_2);((\text{UNION}b\cdot (b\in {\alpha }_2|{({\beta }_S(b\mapsto b);{\eta }_{2,b})}^{-1};{\delta }_{S_1}(b\mapsto b)))\ll {\tau }_2)$

At this stage it should be clear how to calculate this, so we just give the result:

${\delta }_{S_2}(C_1\mapsto C_2)=\{f_1\mapsto \underset{\_}{C_2.f_2},g_1\mapsto \underset{\_}{C_2.g_2}\}$

Putting the parts together:

We can know show the complete total function:

${\delta }_{S_2}=\{$	$(C_1\mapsto C_1)\mapsto \{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\},$
	$(C_1\mapsto C_2)\mapsto \{f_1\mapsto \underset{\_}{C_2.f_2},g_1\mapsto \underset{\_}{C_2.g_2}\},$
	$(C_2\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_2\mapsto C_2)\mapsto \{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\}\}$

Calculating ${\delta }_{S_3}$

There is nothing new for this calculation so we just present the result:

${\delta }_{S_3}=\{$	$(C_1\mapsto C_1)\mapsto \{f_1\mapsto \underset{\_}{C_1.f_1},g_1\mapsto \underset{\_}{C_1.g_1}\},$
	$(C_1\mapsto C_2)\mapsto \{f_1\mapsto \underset{\_}{C_2.f_2},g_1\mapsto \underset{\_}{C_2.g_2}\},$
	$(C_2\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_2\mapsto C_2)\mapsto \{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\},$
	$(C_1\mapsto C_3)\mapsto \{f_1\mapsto \underset{\_}{C_2.f_2},g_1\mapsto \underset{\_}{C_3.g_1}\},$
	$(C_2\mapsto C_3)\mapsto \{f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\},$
	$(C_3\mapsto C_3)\mapsto \{f_1\mapsto \underset{\_}{C_3.f_1},g_1\mapsto \underset{\_}{C_3.g_1},f_2\mapsto \underset{\_}{C_2.f_2},g_2\mapsto \underset{\_}{C_2.g_2}\},$
	$(C_3\mapsto C_1)\mapsto \mathrm{\varnothing },$
	$(C_3\mapsto C_2)\mapsto \mathrm{\varnothing }\}$

Deriving the dynamic binding function

The function is now given. We evaluate it at two interesting points:

• $\delta (C_1\mapsto f_1)=\{C_1\mapsto \underset{\_}{C_1.f_1},C_2\mapsto \underset{\_}{C_2.f_2},C_3\mapsto \underset{\_}{C_2.f_2}\}$

• $\delta (C_1\mapsto f_g)=\{C_1\mapsto \underset{\_}{C_1.f_g},C_2\mapsto \underset{\_}{C_2.g_2},C_3\mapsto \underset{\_}{C_2.g_1}\}$